Mathematical problems involving scenarios with changing rules or rates utilize functions defined in pieces, each applicable over specific intervals. For example, a taxi fare might be calculated based on a starting fee plus a per-mile charge, but the per-mile charge could change after a certain distance. Representing and solving these situations requires constructing and manipulating functions that reflect these varying conditions.
This approach allows for accurate modeling of complex, real-world phenomena in fields ranging from economics and engineering to computer science and physics. Its historical development is closely tied to the broader evolution of calculus and the increasing need to represent discontinuous or segmented processes mathematically. Such segmented functions provide powerful tools for analysis and optimization, enabling more precise and nuanced solutions than simpler, continuous functions often permit.
